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Thèse de doctorat/ PhD Thesis Citation APA:
Capela, F. (2014). Black holes and the dark sector (Unpublished doctoral dissertation). Université libre de Bruxelles, Faculté des Sciences – Physique,
Bruxelles.
Disponible à / Available at permalink : https://dipot.ulb.ac.be/dspace/bitstream/2013/209310/4/778b107b-6ae3-4ccd-aa49-81f424f20470.txt
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D 03963
Black Holes and the Dark Sector
Doctoral Thesis in Theoretical Physics
Fabio Capela
Université Libre de Bruxelles
Service de Physique Théorique
Academie advisor:
Prof. Peter Tinyakov
Deposit: 24 March 2014
Private Defense: 29 April 2014
Public Defense: 20 May 2014
Université Libre de Bruxel es
Black Holes and the Dark Sector
Doctoral Thesis in Theoretical Physics
Fabio Capela
Université Libre de Bruxelles
Service de Physique Théorique
Academie advisor:
Prof. Peter Tinyakov
Deposit: 24 March 2014
Private Defense: 29 April 2014
Prof. Glenn Barnich
Service de Physique Mathématique des Intéractions Fondamentales, ULB, Brussels
Prof. Malcolm Fairbairn
Theoretical Particle Physics and Cosmology Group, Physics Department, King’s College London
Prof. Christophe Ringeval
Center of Cosmology, Particle Physics and Phenomenology, Institute of Mathematics and Physics, Louvain University
Prof. Michel Tytgat
Service de Physique Théorique, ULB, Brussels
A
cknowledgements
It’s time to thank the people who hâve helped me to grow throughout these years. First, I would like to thank my superviser Prof. Peter Tinyakov. During the last four years, I hâve enjoyed ail the physics that I hâve learned from him and the fact that he always has the door open. Peter has been able to guide me, while giving me the freedom to pursue my own research projects. I was able to hâve a very good time during my PhD thanks to Peter, who is always smiling while doing physics. Peter, I don’t hâve many words to say but thank you. You always pushed me to do better. I was lucky to hâve you as a supervisor.I would also like to thank my collaborators Germano Nardini and Maxim Pshirkov. From Germano, I learned to think more deeply about physics and to be meticulous. I hâve also enjoyed the activities that we had during my first year in Brussels, in particular the PES parties. From Maxim, I hâve learned how to do quick estimâtes in astrophysics and that vodka does not give any headache if drunk appropriately. Thanks to Maxim, the discussions in any kind of subject are always lively. This is certainly something that I am missing.
The corridor of the seven floor would certainly be less enjoyable without its Professors. A spécial thank you goes to Prof. Michel Tytgat, Prof. Thomas Hambye and Prof. Jean-Marie Frère for sharing with us their savoir-faire in physics in particular and in life in general. I would like to thank Michel in particular for ail the time that I stole from him during the postdoctoral application and the administrative work. I hâve to say that I enjoy his presence and the discussions that we had so far. His ears are open and I feel that he cares about my future. Thank you. I would also like to thank Prof. Malcolm Fairbairn, Prof. Michel Tytgat, Prof. Christophe Ringeval and Prof. Glenn Barnich for their availability to be part of the jury and their interest on my work.
Of course, I would like to thank ail the past and présent members of the Service: Mikael, Michael, Lorenzo, Germano, Yong, José Mi, Federico, Federica, Simon, Chaimae, Maxim, Laura, Chiara, F\i-Sin, Narendra, Isabelle ... for ail the parties, the discussions and the badminton matches. I will miss you (and I am already missing some of you).
ti Arlindo pelas discussôes sobre o nosso future que esta a tocar a porta. Para terminar, quero dizer um grande obrigado aquela pessoa que me atura todos os dias (e ela sabe que nâo é fâcil). Certamente, esta tese nào teria tido a mesma qualidade sem a tua paciência, carinho e amor. Obrigado a ti, minha amante, amor e amiga.
I don’t know where I will be tomorrow, but I know from where I corne from.
Nâo sou nada. Nunca serei nada.
Nâo posso querer ser nada.
À parte isso, tenho em mim todos os sonhos do mundo.
Fernando Pessoa
I am nothing.
I shall never be anything. I cannot wish to be anything.
Aside from that, I hâve within me ail the dreams of the world.
Short abstract:
This thesis is divided in two parts: the first part is dedicated to the study of black hole solutions in a theory of modified gravity, called massive gravity, that may be able to explain the actual stage of accelerated expansion of the Universe, while in the second part we focus on constraining primordial black holes as dark matter candidates.
In particular, during the first part we study the thermodynamical properties of spécifie black hole solutions in massive gravity. We conclude that such black hole solutions do not follow the second and third of law of thermodynamics, which may signal a problem in the model. For instance, a naked singularity may be created as a resuit of the évolution of a singularity-free State.
In the second part, we constrain primordial black holes as dark matter candi dates. To do that, we consider the effect of primordial black holes when they interact with compact objects, such as neutron stars and white dwarfs. The idea is as follows: if a primordial black hole is captured by a compact object, then the accretion of the neutron star or white dwarf’s material into the hole is so fast that the black hole destroys the star in a very short time. Therefore, observations of long-lived compact objects impose constraints on the fraction of primordial black holes. Considering both direct capture and capture through star formation of primordial black holes by compact objects, we are able to rule out primordial black holes as the main component of dark matter under certain assumptions that are discussed.
To better understand the relevance of these subjects in modem cosmology, we begin the thesis by introducing the standard model of cosmology and its problems. We give particular emphasis to modifications of gravity, such as massive gravity, and black holes in our discussion of the dark sector of the Universe.
C
ontents
1 Introduction 1
1.1 The Big Bang model and its problems... 2
1.1.1 Facing observations: the need for a dark universe .... 4
1.1.2 The challenges... 10
1.1.3 Modifying gravity: the hope for a solution?... 12
1.2 Massive gravity... 13
1.2.1 The vDVZ discontinuity... 15
1.2.2 The Vainshtein mechanism ... 17
1.2.3 Massive gravity: Lorentz invariant & Lorentz breaking . 18 1.3 Black holes... 25
1.3.1 No-hair, thermodynamics and cosmic censorship .... 27
1.3.2 Primordial black holes and dark matter... 31
1
BLACK HOLES AND MASSIVE GRAVITY
35
2 Massive gravity and black hole mechanics 37 2.1 Black hole solutions in massive gravity... 382.2 Properties of the black hole solutions... 41
2.2.1 The black hole température... 41
2.2.2 The Noether’s charge as the entropy... 42
2.2.3 The second and third laws of thermodynamics ... 44
2.3 Discussion... 49
Il BLACK HOLES AND DARK MATTER
51
3 Constraints on black holes as dark matter: direct capture 53 3.1 Capture of black holes by compact stars... 553.1.1 Energy loss... 55
3.1.2 Capture rate... 61
3.2 Constraints... 62
3.2.1 Globular clusters... 62
3.2.2 Dwarf spheroidal galaxies... 63
3.2.3 Results ... 64
3.3 Conclusions... 65
4 Constraints on black holes as dark matter: star formation I 67 4.1 Capture of dark matter during star formation... 68
4.1.2 Adiabatic contraction ... 69
4.1.3 DM bound to a baryonic cloud... 72
4.1.4 Parameters for constraints... 73
4.2 Constraints... 74
4.2.1 Star model ... 76
4.2.2 Sinking of the black hole in the star... 78
4.2.3 Results ... 81
4.3 Conclusions... 82
5 Constraints on black holes as dark matter: star formation II 85 5.1 Adiabatic contraction during star formation... 87
5.2 Constraints on PBHs... 89
5.3 Conclusions... 93
6 Discussion and Conclusions 95
A Parameters and acronyms 99
PREFACE
Our current understanding of the Universe reveals two serious unresolved questions: why is the Universe in a phase of accelerated expansion and what is its matter content. The aim of this thesis is to address these two issues through the study of black holes. Our task is double:
1. Infrared modifications of gravity may potentially account for the accel erated expansion of the Universe without introducing a "dark energy” component. Massive gravity, as we will explain, is a rather interesting candidate for such modified gravity models.
In this thesis, we study the thermodynamical properties of spécifie black hole solutions in massive gravity. If black holes cannot be described as thermodynamical Systems, then it is probable that a UV completion of the low-energy effective theory has rather unusual quantum properties, such as lack of unitarity. Indeed, the second law of thermodynamics, which States that the entropy of an isolated System never decreases, is the resuit of the unitary évolution in quantum mechanics.
Closely related to the thermodynamical properties of black holes is the weak cosmic censorship conjecture, which States that no naked singu- larities can be observed in the Universe, since they are hidden by event horizons. If an infrared modification of gravity predicts the création of a naked singularity from a gravitational collapse, the classical theory becomes inapplicable since quantum effects hâve to be taken into account and a UV completion is needed. We test the weak cosmic censorship con jecture in the framework of massive gravity and discuss the implications of our results for such class of modified théories of gravity.
stars and white dwarfs. The idea is as follows. If a black hole is captured by a compact object, then the accretion onto the hole becomes sufRciently fast to destroy the neutron star or white dwarf in a very short time. As a conséquence, the région of parameter space where this happens with large probability is excluded by observations of the existing compact objects.
The mechanisms of capture that we study are of two sorts: black holes may be captured during the formation of a main-sequence star or directly captured by compact objects. If black holes are captured during the formation of a main-sequence star, like our Sun, nothing happens until the star dies and reaches the stage of a neutron star or a white dwarf. The two mechanisms of capture lead to different and complementary constraints. The reliability of the resulting constraints and the working assumptions will be discussed thoroughly.
Note that the blaek holes discussed in this part are believed to be formed in the early universe, and are, as a conséquence, called primordial black holes. They are usually considered to emerge when the high amplitude of some density fluctuations in the early Universe exceeded some threshold value, leading to an unstoppable collapse.
To better understand the relevance of the previous subjects in modem cos- mology and astrophysics, we first introduce the standard model of cosmology and its problems. We emphasize the rôle that modifications of gravity (in particular massive gravity) and black holes may play for our understanding of the dark sector of the Universe. We also give a detailed discussion about the thermodynamical aspects of black holes and an introduction to primordial black holes and their already existing constraints.
1
I
ntroduction
Somewhere, something incredible is waiting to be known. Cari Sagan
Contents______________________________________________________
1.1 The Big Bang model and its problems... 2
1.1.1 Facing observations: the need for a dark universe ... 4
1.1.2 The challenges... 10
1.1.3 Modifying gravity: the hope for a solution?... 12
1.2 Massive gravity... 13
1.2.1 The vDVZ discontinuity... 15
1.2.2 The Vainshtein mechanisni ... 17
1.2.3 Massive gravity: Lorentz invariant & Lorentz breaking 18 1.3 Black holes... 25
1.3.1 No-hair, thermodynamics and cosmic censorship ... 27
Every single model in physics relies on some working assumptions. In the case of the Big Bang model, one of the axioms is the cosmological principle, meaning the premise that the Universe is spatially homogeneous and isotropie at large scales. The second postulate of the Big Bang model is that the évolution of space-time is described by Einstein’s theory of general relativity (GR) [5]. Based upon those assumptions, a very clear picture of the évolution of the Universe and its constituents has been constructed.
In the next section, we are not aiming to give a précisé picture of the history of the Universe, but rather to présent in words what the standard model of cosmology is telling us. We will fîrst présent a brief history of the Universe, then we will delve into the évidences pointing to the presence of a dark sector in the Universe.
The Big Bang model and its problems
At very high températures, the Universe was in a symmetric phase where the electromagnetic and weak force were the same long-range force, called the electroweak interaction. Moreover, at that point the elementary particles were massless. As the Universe expanded the température dropped. When the température reached a value of the order of 100 GeV (10^^ K), a symmetry- breaking phase transition happened that led the standard model scalar (SMS) to acquire a vacuum expectation value. This had immédiate conséquences: ail the particles that were interacting with the SMS acquired a mass. The second conséquence is that the electromagnetic and weak interaction became two distinct forces, as we conceive them today. This génération of masses of the elementary particles is the well-known Brout-Englert-Higgs mechanism [6, 7, 8]. Since elementary particles acquired a mass, the most massive of them began to annihilate efficiently. First, the top quark annihilated followed by the SMS, then by the gauge bosons W^, Z°.
1.1. The Big Bang model and its problems 3
As the Universe continued to cool down, the weak interaction became weaker and weaker. Eventually, the interaction rate of neutrinos turned smaller than the expansion rate, and they fell ont of thermal equilibrium, meaning they were able to move freely without interactions. This is called the neutrino decoupling. It’s completely similar to the photon decoupling that produces the cosmic microwave background (CMB). Theoretically, it would therefore be possible to observe a cosmic neutrino background that would give us a picture of the Universe when it was two seconds old. However, this cosmic neutrino background unobserved so far is notoriously difficult to detect due to the very weak interactions of neutrinos with matter.
Just after neutrino decoupling, the température of the Universe dropped below 1 MeV, which is the necessary energy to form electron-positron pairs. Therefore, very shortly after the decoupling of neutrinos, the electron-positron annihilation into photons was not compensated by the création of pairs anymore. This implied a huge decrease in the number of électrons and positrons. Moreover, the température didn’t drop as much as before during a short time due to the injection of energy from the electron-positron annihilation.
The photon’s energy decrease with the expansion of the Universe. At some point photons were not able to disrupt deuterium anymore (protons and neutrons tend to produce it easily), since their energy dropped below the binding energy. Consequently, deuterium began to be stable enough time to produce other éléments, like heliura-4. This phase is called the Big Bang nucleosynthesis. As the Universe continued to expand and cool, nuclear fusion processes became inefficient to produce light éléments and the abundances freezed. Today, most of the mass of baryonic matter is in the form of hydrogen (~ 75 %) and hélium (~ 25 %) produced during this epoch. Other much heavier éléments are produced in a much latter phase during the natural évolution of stars, called stellar nucleosynthesis. There is a number of astrophysical observations that can put constraints on the abundances of light éléments at the primordial nucleosynthesis epoch.
Until the Universe was 60’000 years old, radiation was dominating over matter, meaning that most of the energy of the Universe was in the form of photons and neutrinos. Particles like électrons were constantly scattering photons and the Universe was therefore opaque. When the température of the Universe attained the binding energy of hydrogen atoms, free électrons and protons begin to be bound together, letting the photons travel freely in a neutral environment. This is the origin of the CMB that we can measure today.
structures in the Universe, like galaxies and clusters of galaxies. Our présent understanding is that the smallest structures formed first, i.e. stars, through gravitational instabilities. As these objects began to radiate energy, they ionized the plasma (mainly composed of hydrogen) that was neutral at that moment. As the Universe evolved, larger and larger structures were formed: galaxies, clusters and superclusters of galaxies.
As we approach redshift zéro, our Universe begins to accelerate. The cause of the accelerating expansion of the Universe is unknown so far.
Td;>lc 1. 1: Some major eve nts in tlic liistory ol tlie U n i voji'se
Events Température Time
Electroweak phase transition QCD phase transition Neutrino decoupling Electron-positron annihilation Nucleosynthesis Radiation-matter equality Photon decoupling Today 100 GeV 20 ps 150 MeV 20 /rs 1 MeV 1 s 0.5 MeV 10 s 50-100 keV 10 min 0.8 eV BO’OOO yrs 0.3 eV 380’000 yrs 1 meV 14 X 10^ yrs
This is the general picture of the history of the Universe. However, confronted with observations the standard model of cosmology needs two otherwise unde- tected components filling the Universe: dark matter and dark energy. Dark matter is needed to explain some of the astrophysical and cosmological obser vations, while dark energy is mainly required to account for the accelerating expansion of the Universe.
Let us review some of the observations that hâve led the standard model of cosmology to be called nowadays ACDM, A standing for a cosmological constant and CDM meaning cold dark matter.
1.1.1
Facing observations: the need for a dark universe
The Supernovae In 1998, two groups pointed out that the Universe was in a phase of accelerating expansion by observing Type la Supernovae (SN la) [12, 13]. SN la appear when white dwarfs, accreting mass from their binary companions, reach the Chandrasekhar lirait [14] triggering a supernova explosion. Since such a process of supernova explosion is believed to be similar irrespective of where it happens in the Universe, then the absolute magnitude
1.1. The Big Bang model and its problems 5
considered as standard candies and are used to measure astronomical distances. This is possible because of the relation between apparent magnitude m and the luminosity distance di:
The luminosity distance is an important quantity in the previous relation. It is a function of the redshift 2: but it is also dépendent on the density parameter
Cl = p/pc of fluids filling the Universe with équation of state w = pfp, where P corresponds to the pressure of the fluid, p to the energy density and pc to
the critical density required to hâve a spatially fiat Universe. By observing the apparent magnitude and the redshift of the SN la, the two research teams (the Supernova Cosmology Project and High-z Supernova Search Team) were able to deduce independently that the expansion of the Universe is accelerating. To be précisé, the best fit for ail the SN la observations is given by an actual matter density parameter = 0.28lo!os statistical) lo.04 (identified systematics) [12] and a cosmological-constant-like fluid {w = —1) that constitutes around 70% of the energy density of the Universe.
The âge of the Universe Another convincing way to conclude that the Universe is indeed not only filled with matter nowadays is by estimating its âge. Some stars in globular clusters orbiting the Milky Way hâve been estiraated to be 12.7 ± 0.7 Gyr (2o-) old [15]. Therefore, the Universe has to be older than that, i.e. > 12 — ISGyr.
One can make a simple estimate of the âge of the Universe by neglecting the radiation, which is justified since the radiation epoch lasted much less than the âge of the Universe. Doing so, we are able to rule out a Universe only filled with matter, since in that case tu ~ 9.7 Gyr [16], which is less than the âge of the stars in globular clusters. On the other hand, if we consider a spatially fiat Universe with = 0.3, an actual density parameter for a cosmological constant 0^ = 0.7 and the Hubble’s constant from the observations of the Planck satellite [16], then
(1.1) 1 f ~ Ho Jo •00 dz ii + z) [n^(i + 2)3 + fi0]^/^ ~ 14 Gyr, (1.2)
CMB The observations of the CMB hâve separately confirmed the need for an extra contribution in the form of a cosmological constant, totalizing 70 % of the energy density of the Universe.
As we already said, the CMB is extremely homogeneous and isotropie. However, small température fluctuations with observed amplitude of AT/T ~ 10“® are présent across the whole sky [16, 17, 18]. Such fluctuations are considered to hâve their origin in an early phase of the Universe called inflation. They turned out to be a very powerful tool to understand the content, the geometry and the origin of the Universe.
1 1?ab!e 1.2: Cosniolo^ical paraiiieters from Plaiick Cüllal)oraliori [ l.(i] I
Parameter Best fit 68% limits
w 0.022068 0.02207 ± 0.00033
0.6825 0.686 ± 0.020
0.3175 0.314 ± 0.020
i/o (kms“^ Mpc~^) 67.11 67.4 ±1.4
Age/Gyr 13.819 13.813 ±0.058
Before recombination, i.e. when the Universe was filled with électrons and protons that were not bound together, baryons and photons formed a unique fluid with a high rate of interactions between photons and électrons acting as a glue. Two main effects were contributing to the dynamics of this fluid: gravity which compresses the fluid, and photon pressure which is counteracting gravity. Therefore, perturbations, instead of growing like they do after photon decoupling, were undergoing acoustic oscillations. When the photons decou- pled finally from the fluid, the patterns produced by these sound waves were imprinted onto the last scattering surface. By studying the characteristics of the anisotropy spectrum, a rich structure appears. The position and heights of the acoustic peaks give crucial information about cosmological parameters. The first peak corresponds to an acoustic wave that had time to compress once before recombination and is maximally compressed. The higher order peaks hâve gone through several oscillations and are damped with respect to the first one. Even peaks are at maximal rarefication and odd peaks are at maximal compression.
The study of the CMB is a powerful way of constraining the most interesting cosmological parameters. Put together with the SN la observations, we hâve a convincing set of évidences to believe that our Universe is spatially fiat and dominated by a cosmological-constant-like fluid.
1.1. The Big Bang model and its problems 7
matter power spectrum [19, 20]. Useful information can be extracted from this spectrum. In particular, the imprints of the baryonic acoustic oscillations (BAO) hâve been identified in the matter power spectrum [21], Comparing the BAOs at recombination time with its présent value in the matter power spectrum can give useful information about the late-time accelerated expansion of the Universe.
From the CMB observations, something rather strange has emerged; the value of the baryonic density Qb is different from the value of the matter density
Qrn (see Table 1.2). This means that most of the matter in the Universe turns
out to be non baryonic. Such non-baryonic component is called dark matter (DM) and its nature is one of the most intriguing problems in high-energy physics nowadays.
Big Bang Nucleosynthesis Another way to see that Qb doesn’t hâve the same value as the matter density parameter Qm is by considering the Big Bang nucleosynthesis (BBN). The standard model of cosmology offers reliable prédictions for the abondance of the light éléments, D, ®He, '‘He, ^Li. In the Big Bang model, such abondances are directly related to the baryon to photon ratio T] = nbjrij. The computation to obtain the final abondance of deuterium and other éléments reduces to solving coupled Boltzmann équations numerically. The results of the final abondances as a fonction of the baryon to photon ratio
ï] [22] is:
Xz, = 2.60 X 10-^(1± 0.06) , (1.3)
A.u = 4.82 xl0-^»(l± 0.10) . (1.4)
As the abondances are a fonction of 77, getting the observed values of relie abondances serves as a baryometer. To observe the abondances, we need to look at sources with low metallicities to hâve observations as near as possible to the primordial ones.
To infer the primordial abondance of “^He, observations hâve been focused in dwarf galaxies. Based on a considérable amount of data [23], the value
AT4He = 0.249 ±0.009 (1.5)
has been inferred. Deuterium is rather difficult to observe, since it is easily destroyed. However, D has been found in high-redshift quasar absorption Systems [24, 25]. The seven observations hâve led to a value:
nxio'"
Figure 1.1: Relie abundances as a function of the baryon to photon ratio rj. Both the results of the Planck and WM AP collaboration are shown (vertical bands), with the horizontal green bands corresponding to the spectroscopic measurements. The red dot-dashed lines represent the extreme values of the effective number of neutrino familles coming from the Planck collaboration. Taken from [26]
where the error is statistical only. As it is shown in Fig. 1.1, there is a good agreement between prédictions and observations if a value of r; = 5.1—6.5x 10“^° is taken. In the case of the Lithium, observations favour a lower value of t/ compared to the other éléments. This may point to new physics or may simply be related to sources of systematic error that hâve not been taken into account. In any case, observations of metal-poor population II stars of our galaxy give a value [27]:
X7Li = (1.7 ± 0.06 ± 0.44) X 10~^°. (1.7)
1.1. The Big Bang model and its problems
with CMB studies [27]:
0.019 < < 0.024 (95%CL). (1.8)
As a conséquence, we hâve two independent évidences coming from diflFerent epochs that the baryonic density fraction is indeed much less than the matter density. The conclusion is unambiguous: most of the matter in the Universe is non-baryonic. Several observed effects in astrophysical Systems allow us to infer the presence of this non-baryonic component.
Astrophysics There are several observables that are considered to require the presence of dark matter. We will list some of the most compelling of them.
1. Fiat rotation curves: Several observations concluded that the radial ve- locity of stars in galaxies is constant at very large distances from the center of galaxies [28], while Newtonian gravity predicts that stars should hâve a radial velocity that scales with the radius as v{r) oc \fr. To solve this apparent paradox, we may well consider that gravity is modified at large distances [29, 30] (the so-called MOND models) or that extra (dark) matter is contributing much more than the luminous matter at these scales.
2. Lensing: Clusters of galaxies curve spacetime around them so that light emitted by objects behind such clusters travels along curved geodesics and appears several times in our télescopes. The defiection angle of light is related to the mass of the lensing cluster and the impact parameter. By measuring both the impact parameter and the defiection angle one is able to infer that the total mass of the cluster is much larger than the baryonic mass indicated by X-rays from the gas.
This line of reasoning has been applied to the so-called bullet cluster [31]: two colliding clusters where the location of the baryonic gas does not trace the gravitational potential obtained from lensing. This observation is telling us that dark matter is collisionless, contrary to the gas.
3. Hydrostatic equilibrium: after relaxation, a cluster has a gradient of pressure that follows the équation of hydrostatic equilibrium
^ ^ pM(r)pgas(r)
dr
The pressure p{r) oc —ne{r)kBT{r) can be inferred since we can get the température T’(r) from émission lines and the électron number density
Tie(r) from X-ray luminosity. Out of this, the total mass enclosed by a ra dius r can be obtained and compared with the baryonic mass. Once again, the baryonic component is not dominating the matter component [32, 33].
Unless we believe that gravity is modified at large distances, there is a convinc- ing set of évidences pointing to the presence of a non baryonic dark matter component. From what has been said, any dark matter candidate needs to hâve the following properties; it needs to be stable enough to survive until nowadays, “dark”, i.e. interacting (very) weakly with standard model particles, and collisionless, from the bullet cluster observations. Based on N-body simula tions, we are also able to conclude that dark matter should not be relativistic (“hot”) since in that case the top-down scénario is favoured, which States that large structures formed first. Rather, comparison between simulations and observations support a bottom-up scénario [34, 35]. Such a scénario of structure formation is well fitted in simulations considering cold (non-relativistic) dark matter.
1.1.2
The challenges
Dark Matter As long as we consider that gravity is not modified at large distances, we arrive at the conclusion that the Universe is made up of 27% of dark matter. The big question is: what is it? We know that it cannot be baryonic matter and it has to be "dark”, meaning that it doesn’t émit or absorb any electromagnetic radiation at a notable level. Moreover, to hâve a bottom- up formation of structures we need the dark matter to be non-relativistic. There is no particle in the standard model of particle physics that could be dark matter. Consequently, candidates for dark matter usually involve new physics. Examples of popular candidates for dark matter are weakly interactive massive particles (WIMPs), axions, WIMPzillas, Q-balls, stérile neutrinos. A noteworthy counterexample of dark matter candidates that do not involve new physics are primordial black holes (PBHs) and we will corne back to them.
1.1. The Big Bang model and its problems 11
Dark Energy There are two big problems related to the accelerated expan sion of the Universe: the cosmological constant problem and the coincidence problem.
• Cosmological constant problem If we consider that dark energy cor responds to a cosmological constant, then the vacuum energy density needed to fit the data is given by [38];
PA = ~ 10-""GeV^ (1.10)
However, from quantum field theory estimations we would expect a much bigger value for the vacuum energy density. Summing up the zéro point- energies of ail the normal modes of an electromagnetic field, we easily conclude from quantum mechanics that
get an order of magnitude estimate to this theoretical estimation by taking the ultraviolet (UV) cut-off to be of the order of Wmax — 100 GeV, where the electromagnetic interaction is believed to merge with the weak interactions in one single electroweak force. By doing so, we obtain that = lO^^piac^^ Depending on whether we take other UV cut- ofFs, we may alleviate or aggravate such mismatch, but in any case the theoretical prédiction doesn’t match at ail (by many orders of magnitude) the observed value. This is referred as the cosmological constant problem.
• Coincidence problem A rather interesting coincidence is that we are at an âge of the Universe where the matter and the cosmological constant hâve similar abondances. As the Universe expands, we hâve that
Ua PA 3
= — OC a ,
Pm
(1.11)
which implies that at early times the contribution coming from the cosmological constant to the energy density of the Universe was completely negligible compared to the matter contribution. On the other hand, in the future A will take a prédominant rôle. As a conséquence, we are nowadays witnessing a sharp transition between the period where Çl\ was close to zéro and the period where it will be close to 100%. Why is that the case?
1.1.3
Modifying gravity: the hope for a solution?
Ail the previous problems rely on the initial assumptions upon which lies the standard model of cosmology. One of the hypothesis is that the évolution of our Universe is well described by GR. Of course, this is somehow questionable, since the scales where the problems begin to appear are far beyond ail the scales where GR has been tested so far. Therefore, it appears natural to modify gravity at large scales in order to be able to explain the accelerating expansion of the Universe and/or some of the astrophysical observations without any dark matter or dark energy.
As it turns out, some models like Modified Newtonian Dynamics (MOND) [39], or a relativistic version of it [29], are able to explain with high accuracy the galaxy rotation curves without the need to introduce dark matter. More recently, a model that modifies GR has been proposed and daims to reproduce cold dark matter, imitated by a conformai degree of freedom présent in the gravitational sector [40]. As a conséquence, modifications of gravity may provide a plausible explanation to the dark matter problem.
On the dark energy side, an interesting approach to explain the accelerating expansion of the Universe is also to resort to new gravitational degrees of freedom. In particular, in such framework a new low energy scale appears, which is technically natural, i.e. stable to quantum corrections. Such approach ofîers technically challenging problems, and is moreover falsifiable since able to provide spécifie observational prédictions. Among the few interesting examples of modifications of gravity belonging to this category, there are models like DGF, where it is assumed that we résidé on a 3-brane embedded in an infinitely large five-dimensional spacetime [41]. In the DGF model, self-accelerating cos- mological solutions appear due to the 0-helicity component of a five-dimensional graviton that is perceived as a massive résonance [42].
Even though such self-accelerating solutions of the DGF model are plagued by ghost instabilities [43], trying to construct a theory of massive gravitons may still be a viable solution to get rid of dark energy (or dark matter). Indeed, some other examples of théories of massive gravity exist and, as we will see in the next section, are consistent with ail the observations so far. Moreover, unlike DGF models, they hâve their self-accelerating cosmological solution with no instabilities [44].
1.2. Massive gravity 13
graviton produces an infrared modification of gravity, which may be a possible explanation to the dark sector of the Uni verse.
In the next section, we will présent two examples of théories with massive gravitons: a Lorentz invariant model [45] and a Lorentz violating one [46, 47]. The Lorentz invariant model will be shown for completeness and the Lorentz violating massive gravity will be of particular interest for us. Both are self- consistent, healthy and phenomenologically acceptable. More importantly, they may provide a conceivable explanation to the dark sector of the Universe.
Massive gravity
Attempts of constructing a healthy and self-consistent theory of massive gravi tons hâve a long history. The first model of a massive graviton is due to the original work of Pauli and Fierz [48]. They constructed a Lorentz-invariant model of massive gravity in the linear régime with the following action [46]:
S = Mil
J
à^x (^Ceh - ^ - h^)j (1.12)where small fluctuations about a Minkowski spacetime, i.e. +
hfiv, |fipi/[ 1, hâve been considered at the quadratic level in the action. In
the previous équation, £eh corresponds to the Einstein-Hilbert action at the quadratic level in h^i, and h = h^. The Fierz-Pauli action (1.12) has been constructed in such a way that it doesn’t hâve ghosts around the Minkowski background. Indeed, when trying to construct a theory of massive gravity we must be cautions about the propagating degrees of freedom.
+ oth?), where a / — 1, then apart from the 5 degrees of freedom,
we would also hâve an extra scalar mode that would necessarily be a ghost [46], i.e. a kinetic term with a négative sign is présent. Ghosts should be avoided since they correspond to modes with négative energy and induce a vacuum instability, since in their presence the vacuum can decay into ghosts and normal particles. The case a = — 1 of the Fierz-Pauli model is unique, since it gives rise to an extra constraint that kills the ghost, such that the only scalar degree of freedom is the helicity 0 mode with a good sign for the kinetic term.
A simple way of counting the degrees of freedom is to recast the action (1.12) in terms of Hamiltonian variables, i.e. the canonical momentum 7r,j, the spatial slices hij, the temporal component /iqo ^nd the spatio-temporal component hoi of the metric. In this ADM formulation [49], the Lagrangian density in (1.12) takes the form [50]:
Tpp 2/lOi
+ rri^hli + hoo (y^hu - didjhij - rn^hu'j (1-13)
where dot dénotés the time dérivative and the sum over repeated indices is assumed. Moreover, we hâve that
H '^^khijOizhij Oihjf^djhiiç "b dihijdjhjzk
2^ ^ii) * (T14)
From the previous équation (1.13), we see that hoo is appearing linearly multiplying a term that doesn’t hâve time dérivatives, and is therefore a Lagrange multiplier enforcing a constraint C = — didjhij — m?hu = 0. Out of the 12 degrees of freedom (6 of hij and 6 of Tr^) spanning the phase space at each point, the constraint C reduces such number to 11 dynamical degrees of freedom. However, the Hamiltonian H = f d^xTi is not first class and a second constraint arises by imposing that the first constraint is indépendant of time, i.e. the poisson bracket {H,C} ~ 0 vanishes on the constraint surface. Out of the 12 dimensional phase space, the two constraints forces the number of degrees of freedom to 10, the 5 helicity modes of the massive graviton and their conjugate momenta.
On the other hand, if the structure of the mass term is not the one of Fierz-Pauli in (1.12), then hoo appears quadratically in the action and doesn’t impose any constraint. As a conséquence, there are 12 degrees of freedom spanning ail the phase space at each space point, where the extra 2 degrees freedom are related to the ghost and its conjugate momentum.
1.2. Massive gravity 15
case, not only is hoo a Lagrange multiplier but also hoi, since it appears linearly in front of dj^ij and we eliminated the quadratic term in hoi by sending the mass m to zéro. This implies 4 constraints djTTij = 0 and V^hu — didjhij — ha = 0. Moreover, after imposing such constraints, 4 gauge transformations can still be used to eliminate 4 degrees of freedom. Out of the 12 dimensional phase space, only 4 degrees of freedom remain, corresponding to the two polarizations of the massless graviton and their conjugate momenta.
1.2.1
The vDVZ discontinuity
Having a healthy model of massive gravity is good, but at the end of the day one should be able to test it and it should agréé with experiments. In gravity, one of the simplest tests that we can carry out is to measure the strength of interaction between two massive bodies or the angle of deflection of light by a massive body like the Sun. To estimate such interaction, we hâve to couple
the spin-2 massive graviton to matter by a term , where is some
stress-energy tensor. If we vary the action (1.12) with the extra coupling term
, then the équations of motion are [51]:
- hî]^^) + (1.15)
where corresponds to the linearization around the Minkowski background of the Einstein tensor
Sfiw = + ]^dpd^hl
+ -Uh). (1.16)
From Bianchi identities, it turns out that is divergenceless and assuming that the stress-energy tensor is conserved, i.e. d^Tp^, = 0, we obtain by taking the divergence of the équations of motion (1.15):
dPhpp=dph. (1.17)
Such an équation is in fact a constraint that éliminâtes 4 components out of the 10 components of the symmetric metric hp^,. If we take once again a dérivative in the previous équation, we obtain a vanishing linearized Ricci scalar, which when reinserted inside the trace of the équations of motion (1-15) gives a relation for h
We therefore see that the trace of the metric doesn’t propagate and we recover in the absence of sources our previous Hamiltonian analysis with a massive graviton propagating 5 degrees of freedom. Reintroducing the previous two équations (1.17) and (1.18) into the équations of motion (1.15), we are able to obtain an expression for the propagator of a massive graviton, by going to Fourier space. By excluding terms dépendent on momenta, since when contracted with a conserved energy momentum tensor give no contributions, the propagator takes a rather unusual form:
X)("»7^o) ^___ i___ 2 (1.19)
which when compared to the propagator of the massless graviton of GR
'^\ivpa ^ ~ ['Hpp'nav + Vp^Va^p ~ Ipa'Hpi'] i (l-^O) reveals that the third term inside the parenthesis is different from GR even when the graviton mass is vanishing. This is the source of the so-called vDVZ discontinuity [52, 53, 54] (vDVZ standing for van Dam-Veltman-Zakharov).
Such vDVZ discontinuity implies that however small is the mass of the graviton, a subset of prédictions of the Fierz-Pauli model will always differ from the ones of GR. As for example, the gravitational potential of a point source turns out to be 4/3 larger in Fierz-Pauli model than in GR, reflecting the extra contribution from the exchange of the scalar degree of freedom. The vectorial degrees of freedom découplé when the limit m —> 0 is taken. A rather simple way to see this is just to take two sources to which are related two conserved currents and and compute the three level amplitude A for both the massive and massless case. The resuit reads as follows
_^(m=0) ^ y ^ j - \tT'^ ,
For the massive case, we took the large momentum limit. If we take two massive bodies with oc diag(Mi,0,0,0) and T'^‘' oc diag(M2,0,0,0), we
obtain that even when the graviton mass is vanishing. If
ones tries to redefine the Newton constant Gnew = 4/3G, then the discontinuity will reappear in other observables, like the light bending angle which would be 25% smaller than in GR [52, 53, 54], being ruled out by observations [55].
1.2. Mcissive gravity 17
approximation upon which relies the construction of the action (1.12). Even if GR is rather well described by a linear approximation for Solar System scales, it is not the case for the Fierz-Pauli model. Indeed, as the graviton mass approaches zéro, the linear approximation breaks down and non linear interactions hâve to be taken into account. Let’s explain more thoroughly what is happening.
1.2.2
The Vainshtein mechanism
In GR, by constructing spherically symmetric solutions, we are able to isolate the small parameter that Controls the validity of the linearized approximation to be the Schwarzschild radius Tg — 2M/Mpi. Since the metric around the Sun is well described by a spherically symmetric solution, it turns out that the linear approximation is valid only for distances
r»?^~3km. (1.21)
Mp,
The radius of the Sun is much larger than 3 km, and as such, the linear approximation is a good approximation in GR to describe the gravitational field around the Sun. To see if that’s the case for massive gravity or not, we need to hâve a full non-linear theory whose linear expansion around Minkowski background corresponds to the Fierz-Pauli model to guarantee the absence of ghosts. Since the addition of the graviton mass breaks the gauge invariance of GR, a non-linear theory with massive gravitons is not defined uniquely. However, a rather simple example of a non-linear model of massive gravity that approaches the Fierz-Pauli model is just given by a deformation of the fully non-linear GR with the addition of a Fierz-Pauli term [50];
C = (1.22)
corresponds to a fbced metric on which the massive graviton propagates, and therefore the indices of are raised and lowered with it, i.e. g^i, = gj^J + The study of spherically symmetric solutions in this model has first been worked out by Vainshtein [56]. For spherically symmetric solutions, the fixed metric g^J is taken as the Minkowski metric The conclusion was rather surprising.
following form
gf^i.dx^àx'' — —F{r)dt^ + G{r)dr^ + J(r)r^dfî^ (1-23)
an expansion of the type F{r) = Fo{r) + eF\{r) +... was carried ont, revealing that the expansion parameter e — ry/r is singular in the graviton mass m, with
rv (1.24)
being called the Vainshtein radius. Out of this, the conclusion that emerges is that for very small graviton mass, the linear expansion can only be trusted in the région r » 3> Cg, which means that the quadratic action (1.12) cannot accurately describe the Solar System. Indeed, if one takes the value for the graviton mass to be of the order of the actual Hubble scale m 10 eV, which is an interesting value to explain the accelerating expansion of the Universe, then ry ~ lOOkpc for M = M©.
As a conséquence, one may expect that the vDVZ discontinuity is related to the illegitimate use of the linear expansion. Several numerical studies hâve obtained the full non-linear solutions in certain setups, and concluded that non-linearities can indeed restore continuity with GR for r <^ry with solutions entering the linear régime for r ry [57, 58]. In the next section, we présent two non trivial examples of non-linear théories of massive gravity with Lorentz-invariant terms and breaking of Lorentz invariance.
1.2.3
Massive gravity: Lorentz invariant &; Lorentz breaking
Constructing a non-linear theory of massive gravity is rather non trivial, because even if the linear expansion around Minkowski background is the Fierz-Pauli model with no ghosts, it turns out that ghosts generically reappear when non-linearities are taken into account. As an example, let’s take the previous action (1.22) with being the Minkowski metric If we put such an action in terms of the ADM variables, where
/_AT2 + N^N^gij
V
9ijJ
then the Lagrangian density C takes the form
(1.25)
^ = {{7^^^g^J-NC-NiC)
1.2. Massive gravity 19
N, Ni are called the lapse and shift, respectively. The functions N and Ni are
Lagrange multipliers in GR, which enforce the constraints C and C® leading to two real propagating degrees of freedom. Instead, in our non-linear model of massive gravity, the action is not linear in the lapse and shift functions. Therefore, the non-linear action has no constraints or gauge symmetries and propagate 6 real degrees of freedom. As it was argued by Boulware and Deser [59], the extra degree of freedom leads to an Hamiltonian unbounded from below. Given that, the extra mode is generically called the Boulware-Deser ghost.
Prom this example, we see that non-linearities indeed change the constraint structure of the theory, since at linear level the theory (1.22) is just the Fierz- Pauli model with no ghosts. Boulware and Deser considered a rather general class of mass terms and concluded that the ghost is always propagating [60], but they didn’t work out the most general case.
It turns out that a covariant non-linear theory of massive gravity exists [61, 62, 63] and has been proved to be free of Boulware-Deser ghosts [64].
Lorentz invariant massive gravity As we hâve discussed so far, one of the difficulties that we face when constructing a non-linear theory of massive gravity is the presence of a ghost mode. This is linked to the fact that a mass term breaks diffeomorphism invariance, which leads to the lapse N and shift Af* being non-dynamical degrees of freedom enforcing no constraints. Integrating out N and N^ leaves a theory of massive gravitons with 6 propagating degrees of freedom with most probably the sixth degree of freedom being a ghost.
A way of getting rid of the ghost degree of freedom is to construct a mass term such that a constraint remains always présent. Such extra constraint is used to eliminate the ghost mode. In general terms, the mass term is a complex fonction of the lapse N, the shift A'* and hij, Cm{N,N'‘, hij). If it is not possible to solve the équations leading to the direct détermination of the lapse N and shift A®, then there is hope that when plugging back the shift fonction A®(A) in the action a constraint remains. This is équivalent to state that the Hessian
constraint imposes that the first constraint is indépendant of time on the constraint surface, similarly as in the Fierz-Pauli model.
AU the problem reduces to find the good mass term that will indeed hâve det(Hab) = 0, leaving a non-linear theory of massive gravitons with 5 propagat- ing degrees of freedom. In order to construct the most general realization of the non-linear Fierz-Pauli mass term, we need to introduce an extra non-dynamical metric which may be taken to be the Minkowski metric The basic building block of the ghost-free massive gravity model is where the square root matrix is defined by \/g~^f\/g~^f = g^^îxu- We hâve to pay attention to the convention = y/— det(g) that still applies, even though for other cases yfË doesn’t represent the déterminant but rather the square-root matrix.
The most general non-linear massive gravity theory that turns out to be ghost-free is given by [64]
5 =
/
d“xv^ 4 iî-|-2m^ ^a„e„(K) , n=2 (1.28)where the matrix K is defined such that y/g~^f = I -I- K and a2 = 1. In the previous action, we hâve e„(lK) that are polynomials of the eigenvalues of K:
eo(lK) = 1, ei(K) = [K], C2(K) = i ([K]^ - [K]^) ,
e3(K) = i([Kp-3[K][K2]+2[K3]),
C4(K) = ^([K]^-6[K]2[k2] + 3[K2]2 + 8[K][K]3-6[K4]),
efe(IK) = 0 for fc > 4 (1-29)
where the square bracket indicates the trace. Based on the action (1.28), models of bigravity hâve also been constructed [65]. Even though the structure of the action (1.28) may seem complicated, there are only 4 terms and 2 free parameters as and «4.
1.2. Massive gravity 21
However, Lorentz invariant massive gravity is rather difficult to handle, since strong nonlinearities are présent around macroscopie sources, spoiling the per- turbative control of the theory in the solar System [68]. Moreover, several issues in this model still need clarification, such as problems of causality [69], but also the absence of healthy homogeneous and isotropie cosmological solutions [70]. These are among the reasons why we will be focusing on a theory of massive gravity with a breaking of Lorentz invariance, where such problems are not présent.
Lorentz breaking massive gravity There is another interesting class of models of massive gravity that is healthy and that has compelling phenomenological conséquences [46, 47, 71]. Such models are based on a rather audacious hy- pothesis: giving up Lorentz invariance in the gravitational sector. It may seem ludicrous to abandon Lorentz invariance, since it is one of the fundamental ingrédients of the standard model of particle physics, which has been verified with enormous accuracy by several tests [72, 73]. By breaking Lorentz invari ance, different species propagate with different maximum velocities even in fiat space [72]. The extremely tight bounds on the different maximum velocities for the standard model particles [74] can easily be satisfied by not coupling directly the Lorentz breaking fields with the standard model ones, apart from graviton loops.
Breaking Lorentz invariance may even be the good path to produce a theory of quantum gravity, as it has been suggested by the Horava-Lifshitz model of gravity [75]. In such a model, Lorentz invariance is considered to be an emergent symmetry of spacetime at low-energies that doesn’t exist in the UV régime. With that in mind, a model has been constructed that achieves power-counting renormalizability by adding terms with higher order spatial dérivatives of the metric, without introducing higher order time dérivatives, so to préservé unitarity.
Instead of breaking explicitly Lorentz invariance leading to the non conserva tion of the matter stress-energy tensor, it is more judicious to break Lorentz invariance dynamically with fields that hâve équations of motion leaving a conserved matter stress-energy tensor. An explicit example of model that employs such mechanism is the ghost condensate model [76]. In that case, Lorentz symmetry is broken by the time-dependent vacuum expectation value of a scalar field, although the graviton is massless in that case.
by [46, 47]
The Fierz-Pauli model is recovered if we take mg = 0, mf = = m\ = m|.
By doing a (3+1) décomposition, it is rather easy to conclude that the mass term in the tensor sector is given by [46, 47]:
TT stands for transverse and traceless, as usual. By taking into account the kinetic term coming from the Einstein-Hilbert action, we hâve two tensor degrees of freedom propagating with a relativistic dispersion relation + m^, where ni2 is the mass of the tensorial modes. The requirement to hâve no tachyonic instabilities leads to the condition > 0.
Interestingly enough, if we take mi = 0 but generic rrii with f 1, then constraints lead to no propagating modes in the scalar sector, but also in the vectorial sector . This implies that the only propagating modes are the tensorial ones with a mass m2- This leads to two immédiate conséquences: there is no ghosts and no vDVZ discontinuity since both originate from the extra degrees of freedom présent when a mass term is added.
Of course, those conclusions are true as long as we are at the level of the quadratic action with perturbations about the Minkowski background. However, as we saw for the case of the Fierz-Pauli model, the fine-tuning mi = 0 is generically spoiled by non-linear interactions, leading to the reappearance of the Boulware-Deser ghost and the extra degrees of freedom.
There is however a graceful exit to this fine-tuning problem. Instead of imposing some fine-tuned relations, they may just be the conséquence of an unbroken part of the gauge invariance of GR. In this way, we may protect the Lorentz breaking model to become pathological for curved backgrounds, leading to an healthy model of massive gravity. There are several residual symmetries, which are interesting in different contexts. However, the residual symmetry that ensures m\ = 0, while letting the other masses unconstrained, is [46, 47]:
One can think of dynamically breaking Lorentz invariance in order for the graviton to acquire a mass as a Brout-Englert-Higgs mechanism for gravity. We know that the gauge group of GR is the full diffeomorphism invariance related to a massless spin 2 particle, i.e. the graviton. Since in a four dimensional
— (1.31)
1.2. Massive gravity 23
space-time, there are four reparametrizations related to the four coordinates, then each of these symmetries could be broken by the vacuum expectation value of scalar fields which dépend on a particular coordinate. In this framework, the scalar fields correspond to the Goldstone bosons of the Brout-Englert-Higgs mechanism. In Lorentz breaking massive gravity, we hâve therefore four scalar fields whose space-time dépendent vacuum expectation values break the Lorentz symmetry. These fields are minimally coupled to gravity through a dérivative coupling and will be referred to as the Goldstone fields.
To ensure that mi = 0, we may only break some spécifie reparametrization symmetries, while still keeping the residual invariance (1.32). This residual symmetry (1.32) translates into the following symmetry in the Goldstone sector:
+ H*(/) (1.33)
with arbitrary functions E' and where d>“, a = 0,1,2,3 are the Goldstone fields. So we start with a generally covariant theory with extra Goldstone fields 0“. Then, these fields acquire background values that dépend on space-time coordinates, breaking the Lorentz symmetry. Therefore, in this framework Lorentz invariance is broken spontaneously.
By imposing homogeneity and isotropy of space, we require the action to be invariant under SO{3) rotations of the fields, i.e. 0* —>• Aj(^, and under the shift symmetry (f)°{x) + X°‘ with constant A“. The most general action
invariant under the residual symmetry (1-32) and the Euclidean symmetry of 3-dimensional space is given by [46, 47]
where the functions X and are given by
X = A~^g^^‘'d^<iPd,4P, d>^4>^di,<p°d''4Pd^4>° A^X (1.34) (1.35) (1.36)
The first term in the action (1.34) is the usual Einstein-Hilbert term and the second one is a function of the space-time dérivatives of the four scalar fields
4>°‘ minimally coupled to gravity. This model of massive gravity is viewed as a
low-energy effective theory valid below the cutoff scale A, where the graviton mass is of the order m ~ A^/Mpj [46, 47]. As already said, the model (1.34) admits a vacuum solution
that breaks spontaneously the Lorentz symmetry. As expected from the initial requirements, the vacuum possesses rotational symmetry.
Prom nonperturbative Hamiltonian analysis in [77, 78], it has been proved that the model (1-34) doesn’t propagate ghosts and there is a total absence of the vDVZ discontinuity. As a conséquence, the Vainshtein mechanism doesn’t need to be introduced to hâve a phenomenologically viable theory.
It is also interesting to see that the cutoff of the effective theory (1-34) is rather low, A = (mMpi)^/^ ~ (10“^ cm)“^, when compared to the cutoff scale of the ghost-free Lorentz invariant massive gravity (1.28), which is A = (m^Mpi)^/^ ~ (10®cm)“^ [46]. This makes the Lorentz invariant massive gravity theory hard to handle, since quantum corrections are rather important at macroscopie scales [68], but also strong nonlinearities are présent around macroscopie sources, turning inappropriate the use of the perturbation theory in the solar System. Such technical inconvénients disappear for the Lorentz breaking massive gravity.
Prom the phenomenological side, it has been found that the cosmological expansion is driven to an attractor point that, in a certain range of parameters, gives rise to the accelerating expansion of the Universe [79]. More refined studies on the cosmology of the model (1.34) may lead to a better understanding of the recent phase of accelerating expansion of the Universe and may provide some enlightenment about the cosmological constant problem.
To conclude this section on massive gravity, we should point out a rather partic- ular property of the model (1.34): the presence of instantaneous gravitational interactions. Their existence in a model with Lorentz-breaking is relatively easy to understand. In GR, the gravitational potentials are ail instantaneous. However, there is no instantaneous interactions due to a subtle cancellation between them in the graviton propagator. This cancellation does not occur in Lorentz breaking massive gravity, because of the presence of Lorentz breaking fields. Instantaneous gravitational interactions hâve been studied in detail in this model [80].
1.3. Black holes 25
Black holes
As we will see in the next chapters, black holes may lead to interesting insights about models of modified gravity, but are also interesting candidates for dark matter. Before we introduce the main subjects that will be treated in this thesis, we should first define what is a black hole.
One of the unambiguous définitions that is considered in the literature is: a spacetime that contains a black hole is a spacetime with two distinct régions, the interior and exterior of the black hole. Such two régions are causally disconnected, and therefore whatever happens inside of the black hole can ne ver reach an observer which is outside of it, even if he waits an infinité amount of time. A more précisé mathematical définition could be given. However, we will not follow this path. The interested reader can hâve a look at [82] and référencés therein.
Out of this définition of black hole, a question emerges: how to define the event horizon, i.e. the surface that séparâtes the two causally disconnected régions? There are several définitions of the event horizon. However, for the cases that we will be interested in, i.e. static and stationary black holes, it has been shown by Carter [83] and Hawking and Ellis [84], that the event horizon must be a Killing horizon. In order to understand what a Killing horizon is, we first need to define the notion of a Killing field.
If a spacetime has an isometry, i.e a diffeomorphism (coordinate transformation) that leaves the metric invariant, then the metric stays unchanged when moving along a vector field corresponding to the infinitésimal generator of such symmetry {Cçg^i, — 0). Such vector field is called a Killing vector field and satisfies the following équation
^ u'îfi — 0- (1.38)
A Killing horizon is simply a hypersurface H where the Killing vector field is null, i.e. = 0. It is important to point out that not ail Killing horizons correspond to event horizons, an example being the Minkowski spacetime that contains Killing horizons but no event horizons.
very near the center of our galaxy with a period of 15.2 years and a pericenter of 1.8 X 10^^ cm from the center of an object that has the characteristics of a
supermassive black hole [86]. Out of the motion of the star S2, we are able to estimate the mass of the central object to be of the order of 4.1 x 10® Mq [87]. On the other hand, the central object needs to hâve radius (much) less than the pericenter of S2, otherwise the star would be accreted. This leads to the conclusion that the central object called Sagittarius A* has to be a black hole, since no astrophysical entity is known to be able to contain such amount of matter in such a restricted volume. Based on the study of radial velocities of X-ray binaries, several sources of stellar-mass black holes hâve also been observed [88]. Nowadays, black holes are a reality for astrophysicists and this opens an interesting window for the study of gravity in the strong-field régime.
It is believed that existing black holes may hâve very large range of possible masses from 10~®g to lO'^^g (10^M©). Their masses are determined from the way they are formed, but also ail the subséquent history of infalling matter. Certainly, the most common way of forming a black hole is after the évolution of a star. The history of a 25 M© star leads naturally to the création of a black hole. Basically what happens is that when the star, at the end of its life, has exhausted ail of its energy, it collapses. Depending on the mass of the collapsing star, a force can counteract the effect of gravity, leading to the création of objects like white dwarfs or neutron stars. In that case, it is the électron or neutron degeneracy pressure, which is the restoring force in a white dwarf or neutron star, respectively. However, such a pressure can only be effective if the mass of the object is not too high and neutron stars or white dwarfs hâve, as a conséquence, a maximum allowed mass, which for the Ccise of neutron stars is no more than 3 M© [89]. Therefore, if the collapsing star is massive enough (> 20M©), then no force can defeat gravity and a black hole is the only endpoint of the collapsing process. For the case of supermassive black holes, the formation process is still unkown, but several models exist that consider for example the collapse of a cloud of gas in the early Universe or the collapse of a supermassive star that has accreted considérable amount of matter through time, or even mergers of several black holes [90].
1.3. Black holes 27
1.3.1
No-hair, thermodynamics and cosmic censorship
In GR, we know that stationary black holes are characterized by three parame- ters: the mass, the angular momentum and the electric charge [91, 92, 93]. This ’no-hair" theorem means that black holes are described by the Kerr-Newman solutions [94], which for the case of astrophysical black holes simplifies to the Kerr solutions [95] (characterized by a mass and angular momentum), since there is a rapid neutralization of the charge. The no-hair theorem provides a way of testing GR by looking at the geometry of spacetime around astrophysical black holes. Some studies hâve considered to analyse spécifie spectral fines, which are seen in X-ray émissions from stellar-mass and supermassive black holes and are sensitive to the spacetime geometry around the black holes [96]. However, this is difficult since one needs to extract the relevant information that will tell us something about the black hole metric from the effects happening in the accretion disk. Gravitational waves are another way of probing the black hole metric. In particular, the time dependence of the emitted gravitational waves after a merger and with a detailed study of the multipole moments of the black hole may lead to précision tests of the geometry around the black hole. This opens fascinating perspectives about the study of black hole solutions and their possible déviations from GR.
It turns out that Lorentz violating models of massive gravity évadé such no- hair theorem. Indeed, as we previously pointed out, there are instantaneous interactions that are présent in such a class of models. As a conséquence, it is not a surprise that black holes in Lorentz breaking massive gravity are characterized by other parameters than the mass, the electric charge and the angular momentum [97, 98]. This leads to the conclusion that higher multipole moments of the black holes are no more universal in this model and the geometry around these hairy black holes could be tested by gravitational wave observatories or other experiments sensitive to the black hole metric. It is important to point out that testing long distance modifications of gravity using the black hole metric may be rather tricky (and useless when the no-hair theorems [97] hold), and in this sense the Lorentz breaking theory of massive gravity stands up as a promising candidate to be tested.
